Tetrahedra Sharing Volume, Faces Areas, and Circumradius

From the abstract of my note “Tetrahedra Sharing Volume, Face Areas, and Circumradius: A Hedronometric Approach”:

Volume, face areas, and circumradius sometimes determine multiple —even infinitely-many— non-isomorphic tetrahedra. Hedronometry provides a context for unifying and streamlining previous discussions of this fact.

This was my first attempt to solve someone else’s problem with hedronometry. I’m rather pleased at how well hedronometry fit the circumstances, although I believe some of the subsequent polynomial analysis could use some finessing.

The Open Question:

In the case of “doubly-bisohedral” tetrahedra (Section 2.3), is it ever possible to determine four or more non-isomorphic figures? (The best I can do is three.)

Posted 18 July, 2012 by Blue in Hedronometry, Open Question